In a medium saucepan, combine diced peaches, granulated sugar, and 1 cup water. Bring to a boil, reduce heat, and simmer about 5 minutes until sugar is dissolved and peaches are fragrant. Remove from heat, stir in chopped mint leaves, and let steep until cooled, about an hour. Strain through a fine-mesh sieve, pressing on solids to release juices. Discard solids.
Mix the lemonade
In a large pitcher, combine lemon juice (I always start with ½ cup and increase to taste from there), peach-mint simple syrup, and 4 cups cold water, to your personal sweetness/tartness preferences. (This will also depend on the sweetness of your peaches, which can vary widely. Remember that lemonade will further dilute in flavor when served over ice, so it's best to keep the mixture slightly concentrated.) Pour into ice-filled glasses garnished with additional peach slices and a sprig of mint, if desired.
Variations:
Spike it for the adults: Add bourbon, white rum, peach schnapps, tequila, or citrus-flavored vodka, to taste.
Sparkle it: Substitute 4 cups cold sparking water when mixing the lemonade.
Melba it: Add a generous handful of raspberries with the peaches to the hot simple syrup.
Notes
The lemon juice and cold water proportions are easily adjusted to your personal tartness/strength preferences. Lemonade keeps well, covered, in the refrigerator for up to three days. As it sits, the fruit and mint flavors will develop, especially if you've added any additional peach slices or mint sprigs to the pitcher. If making ahead of time, stir before serving.